Given the following rational function:
f(x) = (x^2 + 6x - 8) / (x – 5)
(a) state the domain.
(b) find the vertical and horizontal asymptotes, if any.
(c) find the oblique asymptotes, if any.
Given the rational function:
f(x) = (x^2 + 6x - 8) / (x – 5)
a. Domain
The domain of a rational function is all real numbers minus points where the denominator (x-5) become zero. Here the point to be removed is x-5=0, or x=5.
The answer in interval notation would be:
(-&infin,5)∪(5,∞)
which is essentially all real less x=5.
b. Asymptotes
Vertical asymptotes occur when the denominator becomes zero. There is one such asymptote for f(x).
Hint: this point has been identified in part (a) above.
Horizontal asymptotes are limits of f(x) as x→-∞ or x→∞.
If these limits do not exist, there are no horizontal asymptotes.
Hint: Evaluate Lim x→±∞ and see if the limits exist.
3. oblique asymptotes
Oblique asymptotes exist when the leading term of the numerator divided by the leading term of the denominator yields a linear term (i.e. of the form kx), where k≠0.
Here, the leading term of the numerator is x^2, and that of the denominator is x.
The quotient is therefore x^2/x=x (k=1).
The oblique asymptote is therefore y=x.
Follow link below for the graph:
http://imageshack.us/photo/my-images/221/1307663049.png/
x^2+x-6/x+3
(a) To determine the domain of a rational function, we need to identify any values of x that would make the denominator equal to zero. In this case, the denominator is (x - 5), so the value x = 5 is not allowed, as it would result in division by zero. Therefore, the domain of the function is all real numbers except x = 5.
(b) To find the vertical asymptotes, we need to check if there are any values of x that make the denominator equal to zero, excluding those that are already excluded from the domain. In this case, the only value that makes the denominator zero is x = 5, which is already excluded from the domain. Hence, there is no vertical asymptote in this rational function.
To find the horizontal asymptote, we examine the degrees of the numerator and denominator. In this case, the numerator has a degree of 2 (highest power of x is 2) and the denominator has a degree of 1 (highest power of x is 1). Since the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote.
(c) To find the oblique asymptote, we need to check if the degrees of the numerator and denominator differ by exactly 1. In this case, the degrees differ by 1, so there is a possibility of an oblique asymptote.
To find the oblique asymptote, we perform polynomial long division. Dividing (x^2 + 6x - 8) by (x - 5), the quotient is x + 11 and the remainder is 57. Therefore, the oblique asymptote is the line y = x + 11.
In summary:
(a) The domain of the function is all real numbers except x = 5.
(b) There is no vertical asymptote and no horizontal asymptote.
(c) The oblique asymptote is the line y = x + 11.